1. Introduction

Diamond is a superhard natural material with many excellent optical, physical and mechanical properties [1–5]. The highest thermal conductivity of 2000W MK-1, largest band gap of 5.5eV and broad optical transparency (deep ultraviolet to far infrared), enables it various important promising applications, such as electronic devices [6], heat dissipation [7], nonlinear optics [1,8], NV center [3,9–11], waveguide [12], cutting tools and so on. However, the extremely flat and smooth diamond surfaces are the most fundamental demands of aforementioned application fields [11]. Unfortunately, it is a key challenge to obtain ultra-smooth surfaces, and researchers are still devoting into developing more efficient polishing technologies since last century. For example, since 1920, Tolkowshy et al [13] first introduced the study of diamond grinding and polishing, a variety of polishing technologies have been demonstrated, such as mechanical polishing [14], chemical-mechanical polishing [15], laser polishing [16], high temperature annealing [11,10], ion beam polishing [17–21], reactive ion beam etching [16,22] and OH radicals assisted polishing [23, 24]. Each of these polishing technologies can process diamond surfaces, resulting in varying degrees of improvement [13]. As we all know, the surface topography quality of diamonds is evaluated by roughness values which are measured by Atomic Force Microscope (AFM) and Optical profilometry (OP), because the surface topography is mainly affected by bulges and pits on it. However, current popular roughness measurement techniques, AFM and OP, are limited by their ability in measuring large area of the surface, as well as the accuracy. It is a common sense that the higher the resolution, the higher measurement accuracy, and the higher time consuming. Thus, the current AFM and OP methods cannot comprehensively, objectively, and quickly assess the surface topography of large-area diamonds.

Compare to AFM and OP, scanning electron microscope (SEM) is a quickly and large-scale surfaces image technique which can intuitively give out the surface topography and micro-structure of sample. But limited by its imaging principles, the longitudinal measurement accuracy, such as surface roughness, of SEM is far beyond to satisfy the requirement of surface topography measurement of diamond in electric device and nonlinear optics. Therefore, it well be making great importance to enable SEM high and accurate longitudinal measurement ability by combing popular image processing. In this work, we are committed to combining popular image processing with mathematical statistics to generate a new mathematical model, Surface Topography Quality (STQ), for fast, large area and efficient surface roughness measurement of diamond surface based on two-dimensional SEM images before and after polishing. A quantitative value of Surface Topography Quality Rate (STQR) can be obtained, which is generally used to evaluate the surface topography quality of diamonds. At the same time, we can extract and mark bulges and pits from initial SEM images during the process. Our STQ strategy provides a potential novel direction for the evaluation of diamond surfaces, which may have important influence on promoting the advancement of high-performance diamond devices.

2. Experimental section

Experimental Materials: A single crystal diamond block with the <100 > direction and the dimension of 5mm×5mm×1mm purchases from PrMat (www.prmat.com).

Experimental Equipment: Scanning Electron Microscope (SIGMA-HD, ZEISS) and Atomic Force Microscope (Dimension Icon, Bruker) are applied to characterize the surface topography of diamonds. Ion beam device from ACMEPOLE (www.acmepole.com) is polished and sputtered diamond surfaces at incident angle 20° in the case of 500 eV Ar+ ion.

Experimental Process: First, a single crystal diamond surface is measured by Atomic Force Microscopy (AFM) in twenty-five regions (each area is 2 µm×2 µm in size), and a 250 µm×187 µm area is obtained by SEM. Then, the diamond is placed on the sample stage of the ion beam device and tilted at 20 degrees, a total of 8 polishing times. The average roughness value of twenty-five regions is measured by AFM after each polishing. Finally, the image of the last polishing is obtained by SEM.

3. Results and discussion

Generally, the smooth and flat diamond surfaces mean high-performance of the device, and ion beam polishing technique is a reliable and excellent way to obtain high quality smooth and flat diamond surfaces as the incident angle of ion beam less than 30° [13,25]. For better verification to our STQ method in fast and accurate determining large area roughness of diamond surfaces based on SEM image, we take a careful control experiment with current popular AFM technique. The parameters of the ion beam device are shown in Table S1 (Supplement 1). Figure 1(a) shows AFM measures twenty-five regions with the dimension of 2µm × 2µm at 512 × 512 point resolution on the diamond surface as evenly distributed as possible each time, about 6 hours per measurement time in average. In order to evaluate the surface topography quality objectively and comprehensively, the roughness value of each measurement is the average of roughness values of twenty-five regions. Figure 1(b) shows the change of the roughness value after each ion beam polishing, with a total of eight polishing times and a total time of about 55 hours (9 measurements). We observed that the roughness value of the surface reduced from 0.831nm to 0.316 nm after eight times polishing. We knew that a low standard deviation indicates that the values tend to be close to the mean value, while a high standard deviation indicates that the values are spread out over a wider range. Figure 1(c) and (d) display that the average roughness of the diamond surface was 0.831 nm, with a standard deviation of 0.134 nm. The average roughness of the diamond surface was 0.316 nm with a standard deviation of 0.036 nm after polishing (Fig. 1(e) and (f)). A standard deviation from 0.134 nm to 0.036 nm shows that the surface roughness of diamond surface tends to be stable as the ion beam polishing larger than 5 times.

 

Fig. 1. (a) Schematic of AFM measurement process (Yellow: diamond, Orange: AFM measurement regions, Silvery white: the AFM probe). (b) Change of the roughness value with order number of ion beam polishing, a total of eight times of polishing. The number of 0 refers to before polishing. The number of 1 refers to the first polishing, and so on. (c) AFM roughness image of one of the twenty-five regions before polishing. (d) Frequency histogram and frequent statistical line chart of roughness values before polishing. (e) AFM roughness image of one of the twenty-five regions after polishing. (f) Frequency histogram and frequent statistical line chart of roughness values after polishing.

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As revealed experimentally in above Fig. 1, AFM is a reliable technique to obtain the surface roughness values of diamond surface. However, almost 55 hours’ time consuming in measuring a total area of 900 µm² (225 regions with the area of 2µm × 2µm per region) would be its Achilles’ Heel in the field of fast and large area topography measurement of large-scale diamond surface. As a comparison, we used a fully area surface topography evaluation based on the SEM image of above sample by we developed STQ process. To verify the accuracy of our model in evaluating the surface topography of the sample, both the surfaces before and after ion beam polishing were investigated. The key idea of STQ mathematical model was building the correspondence and relation between the greyscale information of SEM image and surface topography features (such as surface roughness) of the sample, which combining the image processing and mathematical statistics in a reasonable way. The basic processes including: taking the SEM image of sample surface, accurately identifying and collecting the greyscale information of surface from previous SEM image, establishing the connection between greyscale information and surface topography feature by statistical theory, and conversing and outputting the specific topography information of the target surface sample (such as surface roughness), thus we finally can easily obtain the surface topography information of large area surface in a fast, efficient and reliable way. Figure 2 is a schematic display of the key idea and process of our STQ mathematical model.

As shown in Fig. 2, an image was captured by a sensor, which is expressed as a continuous function F(x, y) of two coordinates in the plane by image denoising. In image processing, image digitization means that the function F(x, y) is sampled into a matrix with M rows and N columns, generating M×N pixels. After that, a value of the sampled image F_s (iΔx, jΔy) is expressed as a digital value. The transition between continuous values of the image function (brightness) and its digital equivalent is called quantization. The number of quantization levels should be high enough to permit human perception of fine shading details in the image. Most digital image processing devices use quantization into K equal intervals. If b bits are used to express the values of the pixel brightness then the number of brightness levels is K = 2b. Therefore, we can easily conclude that the finer the sampling (i.e., the larger M and N) and quantization (the larger K), the better the approximation of the continuous image function F(x, y) achieved. Then, we convert the matrix of the brightness level image with b = 8 to a grayscale image which is a numeric matrix with values in the range 0 to1. The histogram is usually the only global information about the image which is available. Histograms are simple to be calculated and are also suitable for fast hardware implementations. Thus, we define the brightness histogram h_F(z) (formula (1)) to provide the frequency of the brightness value z in the image.

 

Fig. 2. The map of Surface Topography Quality (STQ) mathematical model.

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Nowadays, almost all of the technical fields, in some way, are impacted by digital image processing. Although imaging in the electromagnetic spectrum is still the mainstream so far, there are a number of other imaging modalities also pronounced, especially the SEM technology. Generally, a SEM image records the interaction of electron beam and sample, thus the surface information of the sample, which is presented as a grey dot on the phosphor screen [26]. A complete image is formed by a raster scan of the beam through the sample, much like a TV camera. The electrons interact with a phosphor screen and produce light. Thus, we believe that our STQ model works in obtaining surface topography feature of the diamond surfaces based on the corresponding SEM images. Figure 3 shows the results that SEM images of diamond surfaces are solved by STQ mathematical model before and after polishing. The samples in Fig. 3(a) and 3(e) are both 250µm×187µm in size and were obtained by electron microscope imaging. Since the three-dimensional scale can provide a spatial imagination due to the extra Z direction than that of two-dimensional scale. Thus, we first converted the two-dimensional SEM image into a grayscale image, and then generated a three-dimensional topography display of diamond surfaces (Fig. 3(b) and 3(f)) based on preciously obtained brightness values. It can be clearly observed that there are a lot of bulges and pits in unpolished surface. While after several times polishing, the surface becomes flatter and smoother. As we mentioned before, the brightness histogram reflects a global information of the target surface. In Fig. 3(c) and 3(g), the brightness value of diamond surfaces after polishing tends to fluctuate around 0.35, while the brightness value before polishing almost covers the entire brightness value range. Meanwhile, on the basis of the brightness histogram, the mean µ and the variance σ² are obtained to get the normal distribution curve. Afterward, combining the dual-threshold method with Surface Topography Function and adopting formulas ((2) and (3)), Surface Topography Quality Rate α before and after polishing is obtained to be 2.31% and 25.39%, respectively (see Supplement 1). Finally, we extract bulges and pits on the initial SEM image of diamond surface and mark them in red, as shown in Fig. 3(d) and 3(h).

 

Fig. 3. (i) Before polishing. (ii) After polishing (The eighth polishing). (a, e) The SEM image of diamond surface. (b, f)The three-dimensional topography display of diamond surface (Top). The corresponding projection (Bottom). (c, g) The brightness histogram and the normal distribution curve. (d, h) Characterization the surface topography of diamond (Red: extracted bulges and pits).

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To further demonstrate the validity and reliability of our proposed STQ strategy, we process another diamond with same polishing parameters. For simplicity, we polish the surface with a quick time of 20min per time. Figure 4(a) shows the variation of roughness values measured by traditional AFM for four times polishing. It can be seen that the initial surface roughness of the sample, 1.939nm, is flattened to 0.356nm after four times polishing. Consistent with the previous results, when the polishing time is greater than 40 min, the roughness value further decreases and finally tends to be stable. In addition, the 3D topography display of diamond surfaces and brightness histogram are mentioned in Supplement 1. Figure 4(b) and 4(d) are the SEM image of the diamond surface before and after polishing. Surprisingly, as shown in Fig. 4(c) and 4(e), the results of the surface roughness of the same sample analyzed by our proposed STQ mathematical model are perfectly in agreement with that of the results obtained by AFM. The Surface Topography Quality Rate α, Calculated by the STQ mathematical model, decreased from 44.47% (before polishing) to 3.27% (after polishing), indicating the high reliability and validity of our proposed STQ strategy in fast and accurately evaluate the surface quality of diamond. Therefore, the experimental results give out a crystal clear and valid proof to our method in evaluating the surface roughness of large-scale diamond sample with high efficiency and accuracy, as well as low time consuming.

 

Fig. 4. (a) Change of the roughness value with order number of ion beam polishing, a total of four times of polishing. The number of 0 refers to before polishing. The number of 1 refers to the first polishing, and so on. (b) The SEM image of diamond surface before polishing. (c) Characterization the surface topography of diamond before polishing. (Red: extracted bulges and pits). (d) The SEM image of diamond surface after the fourth polishing. (e) Characterization the surface topography of diamond after the fourth polishing. (Red: extracted bulges and pits).

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4. Conclusions

In summary, we demonstrated a novel approach to evaluate large-scale diamond surface quality with high efficiency and accuracy, which is titled as STQ mathematical model by combining understandable image processing techniques and the knowledge of mathematical statistics. Compared with traditional AFM, SEM images, the proposed strategy enables a larger measurement area and a shorter time cost. Besides, we qualified the surface topography quality of large-area diamonds by Surface Topography Quality Rate, which is not only digitizing the diamond surface quality, but revealing its three-dimensional morphology. We believe that the work we demonstrated in this paper will provide new guidance for evaluating the quality of large-area diamond surfaces, and may indirectly promote various applications of diamond.